Embedded newform 310.3.i.a.99.13

• Label: 310.3.i.a.99.13
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			99.13
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.29

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(1.41600 + 2.45259i) q^{3} -2.00000 q^{4} +(4.35084 - 2.46378i) q^{5} +(3.46848 - 2.00253i) q^{6} +(-0.711149 + 0.410582i) q^{7} +2.82843i q^{8} +(0.489879 - 0.848494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.41421i		−	0.707107i
							\(3\)	1.41600	+	2.45259i	0.472001	+	0.817529i	0.999487	−	0.0320347i	\(-0.0101987\pi\)
−0.527486	+	0.849564i	\(0.676865\pi\)
\(5\)	4.35084	−	2.46378i	0.870167	−	0.492756i
							\(7\)	−0.711149	+	0.410582i	−0.101593	+	0.0586546i	−0.549936	−	0.835207i	\(-0.685348\pi\)
0.448343	+	0.893862i	\(0.352014\pi\)
\(11\)	−6.49906	−	3.75223i	−0.590824	−	0.341112i	0.174599	−	0.984640i	\(-0.444137\pi\)
							−0.765423	+	0.643527i	\(0.777470\pi\)
\(13\)	12.5415	−	21.7226i	0.964733	−	1.67097i	0.254402	−	0.967099i	\(-0.418121\pi\)
							0.710331	−	0.703868i	\(-0.248545\pi\)
\(17\)	7.77837	+	13.4725i	0.457551	+	0.792502i	0.998831	−	0.0483409i	\(-0.0153934\pi\)
							−0.541280	+	0.840842i	\(0.682060\pi\)
\(19\)	11.9347	+	20.6716i	0.628144	+	1.08798i	0.987924	+	0.154940i	\(0.0495184\pi\)
							−0.359780	+	0.933037i	\(0.617148\pi\)
\(23\)	12.4457			0.541118			0.270559	−	0.962703i	\(-0.412791\pi\)
							0.270559	+	0.962703i	\(0.412791\pi\)
\(29\)		−	16.3903i		−	0.565183i	−0.959240	−	0.282592i	\(-0.908806\pi\)
							0.959240	−	0.282592i	\(-0.0911941\pi\)
\(31\)	−30.9975	−	0.395949i	−0.999918	−	0.0127725i
							\(37\)	12.9346	+	22.4033i	0.349583	+	0.605496i	0.986175	−	0.165705i	\(-0.0529899\pi\)
−0.636592	+	0.771201i	\(0.719657\pi\)
\(41\)	8.35237	−	14.4667i	0.203716	−	0.352847i	−0.746007	−	0.665938i	\(-0.768031\pi\)
							0.949723	+	0.313092i	\(0.101365\pi\)
\(43\)	0.465143	+	0.805650i	0.0108173	+	0.0187361i	0.871383	−	0.490603i	\(-0.163223\pi\)
							−0.860566	+	0.509339i	\(0.829890\pi\)
\(47\)		−	9.31396i		−	0.198169i	−0.995079	−	0.0990847i	\(-0.968409\pi\)
							0.995079	−	0.0990847i	\(-0.0315915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.13	✓	64
5.4	even	2	inner	310.3.i.a.99.20	yes	64
31.26	odd	6	inner	310.3.i.a.119.4	yes	64
155.119	odd	6	inner	310.3.i.a.119.29	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.13	✓	64	1.1	even	1	trivial
310.3.i.a.99.20	yes	64	5.4	even	2	inner
310.3.i.a.119.4	yes	64	31.26	odd	6	inner
310.3.i.a.119.29	yes	64	155.119	odd	6	inner